[/quote]
,
Prove that
and 
.![$$ a^2-ab+b^2 \geq \sqrt[3]{2} $$](http://latex.artofproblemsolving.com/7/9/8/798484c6b985ebbace2a3dde816f11a46635feba.png)
Let 
.![$$ 3\geq a^2+ab+b^2 \geq \sqrt[3]{4} $$](http://latex.artofproblemsolving.com/d/7/7/d77fbdc50a88395259e58d31413b9ac453e5dfd0.png)
of odd degree with real coefficients such that 
.
be non-negative real numbers such that 
Prove that
be an acute triangle with 
and 
.
be the midpoint of the side 
.
intersects the side 
again at 
and the circumcircle of triangle 
intersects the side 
again at 
.
lies on the perpendicular bisector of the segment 
lie on the same side of 
and 
.
and 
are tangent.
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino.
Note: 
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
be real numbers less than 2.Then prove that 
with 
Prove that
)
.
passes through 
and 
.
and 
,
be the midpoint of 
.
,
and 
be a quadrilateral such that the excircles 
and 
of triangles 
touching their sides 
respectively touch the extension of 
and 
.
and 
is right
such that 
has a divisor in the range 
.
and no elements of 
and 
,
is an integer.
.
real numbers 
and 
.
of real numbers for which 
,
are pairwise distinct. For every such pair sort the corresponding values 
increasingly and let 
be the 
-
.
.
Y by Mualpha7, orl, ivanbart-15, vntbqpqh234, AkshajK, Love_Math1994, sinhaarunabh, arqady, TheLittleOne, goodar2006, hsiljak, mathway, Rijul saini, VIPMaster, vanthanh0601, Mrdavid445, Goutham, hh1546, professordad, BigSams, hyperspace.rulz, shinichiman, ACCCGS8, WakeUp, euler_brick, ck204, littlejones, crycry-tara1995, Teodor95, Mateescu Constantin, Nguyenhuyhoang, bluecarneal, Potla, shivangjindal, Fatlummi, dien9c, pi37, seby97, AnhIsGod, 61plus, Algie, NewAlbionAcademy, Binomial-theorem, iarnab_kundu, mlm95, cooper, bboypa, bappa1971, tim9099xxzz, bryanbr2, SWARNAKAMAL, applepi2000, Sampro, Draco, alibez, joshxiong, El_Ectric, feri_AI, mentalgenius, clonewarvb, claudiafeng, Ragnarok7, airplanes1, MinatoF, mathmdmb, underzero, Sayan, siddigss, BOGTRO, aZpElr68Cb51U51qy9OM, NaeunLove, mathman523, antimonyarsenide, borntobeweild, mahanmath, helya, sina, GinMellkior, dan23, zabihpourmahdi, dantx5, fermat007, Isogonics, Pinphong, mathman500, va2010, ShahinH, baijiangchen, msop, chocolate9, Magnolia, etude, guilt, Lord.of.AMC, mathgenius64, yugrey, r31415, negativebplusorminus, ProblemSolver1026, mariekitty, fortenforge, asf, ssilwa, Super, JigglykongisFUM16, Tungsten, ScottBusche, jeremylu, 54math, aquakitty11, ThinkFlow, MSTang, jeffchen, ninjataco, infiniteturtle, yankeesfan, 155919, forthegreatergood, bcp123, MathLearner01, msaeids, HYP135peppers, tastymath75025, thkim1011, Soccer_Girl, fractals, DiscipulusBonus, AdvitiyaBrijesh, acupofmath, kevinkore3, Mathematicalx, misgiven, Klaus-Anton, champion999, SHARKYBOY, TheCrafter, laegolas, Superwiz, SirCalcsALot, irishfeet123, TRYTOSOLVE, minimario, QuadraticReciprocity, mrowhed, Gibby, Einstein314, Magikarp1, tarzanjunior, ESCHen99, mutasimmim12, eisirrational, nmd27082001, TomMarvoloRiddle, DrMath, Salty_Titanium, Mathuzb, rodamaral, Giffunk, integrated_JRC, Tawan, QWERTYphysics, Supercali, enhanced, Wizard_32, Kagebaka, Fermat_Theorem, Jerry37284, mathleticguyyy, Kamran011, fjm30, hansduran0123, duwipr, trololo23, Chipaoo, guavaSeabird, Shinichi-123, huashiliao2020, mathmax12, postingunsolvedquestions, EpicBird08, alephnull, Adventure10, and 12 other users
After a long while, I'm back again.
But this time, with a really huge collection of problems...
I didn't want to share this since it was the source of the book I'm writing, but now I think this is not a good reason...
Dedicated to all AoPS users.
PS. This was a boring work of mine for about one week. All the problems have the links in the forum and I have mentioned them all in the PDF file. I edited the first 600 problems in order to fix the LaTeX/Math errors of this file. Please, if you can help me edit this huge collection, send me a PM.
But this time, with a really huge collection of problems...
I didn't want to share this since it was the source of the book I'm writing, but now I think this is not a good reason...
Dedicated to all AoPS users.
PS. This was a boring work of mine for about one week. All the problems have the links in the forum and I have mentioned them all in the PDF file. I edited the first 600 problems in order to fix the LaTeX/Math errors of this file. Please, if you can help me edit this huge collection, send me a PM.
i
Thank you.
and 
are positive integers satisfying 
.
.
